Someone requested a hypercube which could be manually rotated so I threw this together. Drag any of the sliders to rotate the hypercube around each of the six planes of rotation. You may notice that some rotations seem to have no effect when seem in orthographic or 3D perspective because the rotation is happening in directions which are not represented in those views when seen from some angles.
A hypercube (or tesseract) is the 4-dimensional analogue of a cube. It has 16 corners, 36 edges, and 24 sides. All corners are right angles although they may seem like acute or obtuse angles when seem from a different viewpoint, just as the corners of a cube may seem sharper or wider depending on your viewpoint.
Rotation is achieved through a simple 4x4 transformation matrix which allows the hypercube to be rotated along a single plane at a time. In 3-dimensional rotation, rotation usually occurs around a linear axis but 4-dimensional rotation occurs around a planar axis.
You can check/uncheck the box to see the hypercube with solid sides or empty sides. Click on "Perspective" to select different methods of display.